| L(s) = 1 | − 1.80·2-s − 3.34·3-s + 1.24·4-s + 2.67·5-s + 6.02·6-s + 1.66·7-s + 1.36·8-s + 8.19·9-s − 4.80·10-s − 2.09·11-s − 4.15·12-s + 2.30·13-s − 3.00·14-s − 8.93·15-s − 4.94·16-s + 3.16·17-s − 14.7·18-s + 0.855·19-s + 3.31·20-s − 5.58·21-s + 3.76·22-s + 23-s − 4.56·24-s + 2.13·25-s − 4.15·26-s − 17.3·27-s + 2.07·28-s + ⋯ |
| L(s) = 1 | − 1.27·2-s − 1.93·3-s + 0.620·4-s + 1.19·5-s + 2.45·6-s + 0.631·7-s + 0.482·8-s + 2.73·9-s − 1.52·10-s − 0.630·11-s − 1.19·12-s + 0.640·13-s − 0.803·14-s − 2.30·15-s − 1.23·16-s + 0.768·17-s − 3.47·18-s + 0.196·19-s + 0.741·20-s − 1.21·21-s + 0.803·22-s + 0.208·23-s − 0.932·24-s + 0.427·25-s − 0.815·26-s − 3.34·27-s + 0.391·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5381294151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5381294151\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.80T + 2T^{2} \) |
| 3 | \( 1 + 3.34T + 3T^{2} \) |
| 5 | \( 1 - 2.67T + 5T^{2} \) |
| 7 | \( 1 - 1.66T + 7T^{2} \) |
| 11 | \( 1 + 2.09T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 19 | \( 1 - 0.855T + 19T^{2} \) |
| 31 | \( 1 + 7.33T + 31T^{2} \) |
| 37 | \( 1 - 6.32T + 37T^{2} \) |
| 41 | \( 1 + 0.237T + 41T^{2} \) |
| 43 | \( 1 - 9.69T + 43T^{2} \) |
| 47 | \( 1 - 6.83T + 47T^{2} \) |
| 53 | \( 1 - 2.54T + 53T^{2} \) |
| 59 | \( 1 - 9.30T + 59T^{2} \) |
| 61 | \( 1 - 5.36T + 61T^{2} \) |
| 67 | \( 1 + 4.05T + 67T^{2} \) |
| 71 | \( 1 + 9.77T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 5.24T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 4.00T + 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.50386089005947023868524394002, −9.869301325897092334162175307032, −9.086674150732245415477722368406, −7.78388126463651227016204111730, −7.06288111689119386243896009407, −5.87654157783093248393195313371, −5.46859973314271628433655245980, −4.38270100565685081078754332270, −1.86579903455996398479064390585, −0.887933692681204713308516191209,
0.887933692681204713308516191209, 1.86579903455996398479064390585, 4.38270100565685081078754332270, 5.46859973314271628433655245980, 5.87654157783093248393195313371, 7.06288111689119386243896009407, 7.78388126463651227016204111730, 9.086674150732245415477722368406, 9.869301325897092334162175307032, 10.50386089005947023868524394002
